EX 3.16 continued table (3.6)

• For the pmf of $X=$ number of courses in Table 3.6:
  • The population mean $\mu$ is calculated as: $$\begin{array}{rl}\mu & =1\cdot p(1)+2\cdot p(2)+\cdots +7\cdot p(7)\\ & =(1)(0.01)+2(0.03)+\cdots +(7)(0.02)\\ & =0.01+0.06+0.39+1.00+1.95+1.02+0.14\\ & =4.57\end{array}$$

Interpretation:

• $\mu =4.57$ is the population mean:
  • This reflects the weighted average of the possible $X$ values $1,2,\dots ,7$, not just a simple average.
  • The weights (probabilities) place more emphasis on certain values, particularly 4, 5, and 6, leading to a mean higher than the ordinary average of $1,\dots ,7$ (which is 4).
• Thus, $\mu =4.57$ reflects the distribution of the number of courses more accurately than just the midpoint of the range.